Chapter 11 Pooling correlation coefficients

To pool correlation coefficients Fishers Z transformation is used. The following formulas are used (Raghunathan (2016), Van Buuren (2018) and C. K. Enders (2010)):

\[\begin{equation} Z_i = \frac{1}{2}ln\frac{1+r_i}{1-r_i} \tag{11.1} \end{equation}\]

The \({Z_i}\) means the calculation of Fisher’s Z-value in each imputed dataset.

Also, the variance of the correlation can be calculated using:

\[\begin{equation} Var_Z=\frac{1}{n-3} \tag{11.2} \end{equation}\]

n is the sample size in the imputed dataset. Now we can use Rubin’s Rules to calculate the Pooled correlation and variance. These values will be calculated with the transformed Z values.

To obtain the pooled p-value for the correlation coefficient we use the formula:

\[\begin{equation} Z=\frac{Z_{Pooled}}{\sqrt{Var_Z}} = \frac{Z_{Pooled}}{\frac{1}{\sqrt{n-3}}}=Z_{Pooled}\times\sqrt{n_i-3} \tag{11.3} \end{equation}\]

In this formula z is the z-score and follows a standard normal distribution, \(Z_{Pooled}\) is the pooled Z transformation and \(Var_Z\) is the pooled variance.

Finally, back transformation to the original scale of r is done by:

\[\begin{equation} r_{Pooled} = \frac{e^{2\times\\Z_{Pooled}}-1}{e^{2\times\\Z_{Pooled}}+1} \tag{11.4} \end{equation}\]

References

Raghunathan, T. 2016. Missing Data Analysis in Practice. Boca Raton, FL: Boca Raton: CRC Press.

Van Buuren, S. 2018. Flexible Imputation of Missing Data. Second Edition. Boca Raton, FL: Chapman & Hall/CRC.

Enders, Craig K. 2010. Applied Missing Data Analysis. Guilford Press.